1. ## homeomorphism help

Hi, learning about homemorphic functions at the moment and am failry confused,

If i have a homeomorphic function h: A - B between two topological spaces A and B,

Is the pre image of the empty set in B the empty set in A ?

2. ## Re: homeomorphism help

let me ask you this: can the pre-image of the empty set in B have ANY elements in it?

3. ## Re: homeomorphism help

Is it that it cant because a homeomorphic function is bijective and continuous?

4. ## Re: Homeomorphism help

Look closely at the definition of pre-image: If $\displaystyle f:A\to B$ and $\displaystyle V\subseteq B,$ the pre-image of $\displaystyle V$ under $\displaystyle f$ is $\displaystyle f^{-1}(V)=\{x\in A:f(x)\in V\}.$ In other words, $\displaystyle x\in f^{-1}(V) \Leftrightarrow f(x)\in V$ for all $\displaystyle x\in A.$ Now try assuming $\displaystyle f^{-1}(\O)\ne\O$ and see what happens.

5. ## Re: Homeomorphism help

i'm sorry i understand the definition but i think i may have some understanding of the empty set missing, obviously this cant happen but why cant a function map something to the empty set and the empty set to something else

6. ## Re: Homeomorphism help

Originally Posted by monster
why cant a function map something to the empty set and the empty set to something else
We're not talking about the empty set as an element of $\displaystyle A$ or $\displaystyle B;$ we're talking about it as a subset of $\displaystyle A$ and $\displaystyle B.$